## Newton's three-body problem explained - Fabio Pacucci

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In 2009, two researchers ran a simple
experiment.
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They took everything we know about our
solar system and calculated
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where every planet would be up to 5
billion years in the future.
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To do so they ran over 2000 numerical
simulations
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with the same exact initial conditions
except for one difference:
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the distance between Mercury and the Sun,
modified by less than a millimeter
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from one simulation to the next.
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Shockingly, in about 1 percent of their
simulations,
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Mercury’s orbit changed so drastically
that it could plunge into the Sun
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or collide with Venus.
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Worse yet,
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in one simulation it destabilized
the entire inner solar system.
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This was no error; the astonishing variety
in results
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reveals the truth that our solar system
may be much less stable than it seems.
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Astrophysicists refer to this astonishing
property of gravitational systems
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as the n-body problem.
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While we have equations that can
completely predict the motions
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of two gravitating masses,
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our analytical tools fall short when
faced with more populated systems.
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It’s actually impossible to write down
all the terms of a general formula
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that can exactly describe the motion
of three or more gravitating objects.
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Why? The issue lies in how many unknown
variables an n-body system contains.
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Thanks to Isaac Newton, we can write
a set of equations
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to describe the gravitational force
acting between bodies.
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However, when trying to find a general
solution for the unknown variables
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in these equations,
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we’re faced with a mathematical
constraint: for each unknown,
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there must be at least one equation
that independently describes it.
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Initially, a two-body system appears to
have more unknown variables
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for position and velocity than
equations of motion.
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However, there’s a trick:
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consider the relative position and
velocity of the two bodies
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with respect to the center of
gravity of the system.
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This reduces the number of unknowns
and leaves us with a solvable system.
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With three or more orbiting objects in the
picture, everything gets messier.
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Even with the same mathematical trick
of considering relative motions,
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we’re left with more unknowns than
equations describing them.
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There are simply too many variables
for this system of equations
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to be untangled into a general solution.
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But what does it actually look like for
objects in our universe
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to move according to analytically
unsolvable equations of motion?
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A system of three stars––
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like Alpha Centauri could come crashing
into one another or, more likely,
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some might get flung out of orbit
after a long time of apparent stability.
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Other than a few highly improbable
stable configurations,
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almost every possible case is
unpredictable on long timescales.
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Each has an astronomically large range
of potential outcomes,
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dependent on the tiniest of differences
in position and velocity.
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This behaviour is known as chaotic
by physicists,
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and is an important characteristic
of n-body systems.
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Such a system is still deterministic—
meaning there’s nothing random about it.
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If multiple systems start from the exact
same conditions,
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they’ll always reach the same result.
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But give one a little shove at the start,
and all bets are off.
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That’s clearly relevant for human space
missions,
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when complicated orbits need to
be calculated with great precision.
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in computer simulations
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offer a number of ways
to avoid catastrophe.
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By approximating the solutions with
increasingly powerful processors,
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we can more confidently predict the motion
of n-body systems on long time-scales.
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And if one body in a group of three is so
light it exerts no significant force on the other two,
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the system behaves, with very good
approximation, as a two-body system.
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This approach is known as the “restricted
three-body problem.”
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It proves extremely useful in describing,
for example,
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an asteroid in the Earth-Sun
gravitational field,
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or a small planet in the field of a
black hole and a star.
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As for our solar system, you’ll
be happy to hear
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that we can have reasonable confidence
in its stability
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for at least the next several
hundred million years.
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Though if another star, launched from
across the galaxy, is on its way to us,
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all bets are off.
Title:
Newton's three-body problem explained - Fabio Pacucci
Speaker:
Fabio Pacucci
Description:

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Video Language:
English
Team:
closed TED
Project:
TED-Ed
Duration:
05:09
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